I have the following PROBLEM IN LINEAR ALGEBRA, I do not know the answer.
Assume that d and n are natural numbers and define $f: \R^d \to \R$ by
$$
f(x) = (\prod_{l=1}^d \cos^2(x^l)) - 1/n ,
$$
where $x=(x^1,...,x^d)$. Hence $x^l$ is the $l$th component
of the vector $x$. Prove or disprove the following CONJECTURE:
For any given $x_1,...,x_n \in \R^d$ the $(n,n)$-matrix
$A$ given by
$$
a_{ij} = f(x_i-x_j)
$$
is positive semidefinite, i.e., the eigenvalues are nonnegative.
(Comment: I know that this is true for $n \ge 2^d$. So the
interesting case would be $n < 2^d$.)

CA-TEACH is an internet mailing list devoted to the discussion of the teaching
of COMPLEX ANALYSIS. Though it is intended to be a list devoted to teaching
in an undergraduate setting, it is not restricted to this.

Regular topics on CA-TEACH could include: a discussion of Complex Analysis
textbooks, broadcasts of URLs where documents on complex analysis can be
found, personal reports on teaching complex analysis courses at your
institution, impact of technology on the teaching of complex analysis,
curriculum debates, lecture notes and examples of using various software
packages to assist in complex analysis visualization, among others.

CA-TEACH will *not* be a moderated list. If people are interested in a
moderated list then I will gladly accept volunteers to be the moderator.

To join the CA-TEACH mailing list, send a message to MAJORDOMO@abacus.oxy.edu
with one line in the message body

SUBSCRIBE CA-TEACH

Any other questions about the list should be sent to
CA-TEACH-approval@abacus.oxy.edu or Majordomo-Owner@abacus.oxy.edu

It seems this anniversary will not be celebrated in conferences and
journals, so before it passes I am commemorating the:

50th Birthday of Modern Numerical Analysis
November 1947 -- November 1997

When computers were invented, John von Neumann saw the accuracy
of calculations would need study. He made an example of Gaussian
elimination because many people thought it would be unsuitable for
automatic computing. At Princeton, in the winter of 1946-47, von
Neumann proved the computer he was building could invert matrices
larger than thought possible at the time. The proof appeared in
"Numerical Inverting of Matrices of High Order" [Bull AMS, Nov '47]
written with Herman Goldstine. Few people read the paper in detail,
but some who mattered did. Wallace Givens and James Wilkinson
adopted the paper's approach, and made the paper the model for
rounding error analysis (alternatives now include automatic,
interval and statistical analyses).

Although many parts of numerical analysis existed before von
Neumann's paper, they coalesced as an academic discipline in the
subsequent decade. Von Neumann and Goldstine's paper has been
called the first in this "modern" numerical analysis because it is
the first to study rounding error and because much of the paper is
an essay on scientific computing (albeit with an emphasis on
numerical linear algebra). The list of error sources in Chapter 1
is clearer and more authoritative than any since. The axiomatic
treatment of rounding error in Chapter 2 inspired later analyses.
The discussion of linear algebra in Chapters 3 to 5 contains
original material, including the invention of triangular factoring.
(Von Neumann is the last of three inventors. The LDU factorization
in particular should be named for him.)

The rounding error analysis in Chapter 6 accounts for just
one-quarter of the paper, with the analysis of triangular factoring
a fraction of that. Von Neumann shows, for symmetric positive
definite matrices, the backward error of factoring equals the sum of
the errors incurred on the Gaussian elimination steps. In the many
years following the paper, no other condensation of the errors has
been proposed. Von Neumann bounds the backward error using this
sum. The bulk of Chapter 6 then bounds the residual of inverting
symmetric positive definite matrices. General matrices are treated
by applying the preceding to something like the normal equations.

Textbooks do a great disservice when they note --- without
explaining why --- von Neumann proved backward error bounds for
factoring positive definite matrices. As a result, many people are
unaware this is von Neumann's discovery and not his oversight. It
is difficult to learn the truth because the matter is not discussed
plainly, but there seem to be only two predictive ("a priori" in the
subject's jargon) backward error bounds. For matrices of order n,
the bound for von Neumann's positive definite case varies by n,
while the bound for the general case varies by 2**n. Von Neumann
even remarked he discovered the positive definite case because he
could not find "satisfactory" error bounds for factoring general
matrices. Small, predictive bounds are the only proof an algorithm
is consistently accurate. To explain things plainly, after 50 years
of work, the only acceptable bound we have for the most widely used
algorithm is the bound we started with, von Neumann's.

The concluding Chapter 7 interprets the rounding error analysis.
Von Neumann asked his readers to continue from his residual bound
"several different ways". He guided his readers by explaining the
appropriate conclusion shows the computed result is exact for some
perturbation of the initial data. This is the "backward"
interpretation which many people think von Neumann did not
understand. The paper closes by evaluating the residual bound for
"random" matrices, and by counting arithmetic operations.

In sum, von Neumann's paper contains much that is unappreciated
or at least unattributed to him. The contents are so familiar, it
is easy to forget von Neumann is not repeating what everyone knows.
He anticipated many of the developments in the field he originated,
and his theorems on the accuracy of Gaussian elimination have not
been encompassed in half a century. The paper is among von
Neumann's many firsts in computer science. It is the first paper in
modern numerical analysis, and the most recent by a person of von
Neumann's genius.

Happy Birthday and Best Regards from Joe Grcar

ps 1. Klara von Neumann named a puppy after her husband's matrix
inversion project. John, Klara and a fully grown Inverse can be
seen in "Passing of a Great Mind", Life, Feb. 25, 1957. Also note
the photo of von Neumann by the famous Time-Life photographer Alfred
Eisenstaedt.

ps 2. Wilkinson's "von Neumann Lecture" [SIAM Rev, '71] takes
passages of von Neumann's paper out of context and recommends
obvious improvements. This has been interpreted to mean the paper
is somehow flawed. As Paul Halmos explained, von Neumann many times
gave lesser mathematicians the opportunity to "improve" von Neumann.

ps 3. Von Neumann and Goldstine's four errors of scientific
computing are these. Last is precision: no computing device can
perform all its operations exactly, and when an imperfect operation
is performed, it injects noise into the calculation. Third is
approximation: the formulas of scientific theories must be made
amenable to evaluation by machine operations, and unending searches
for results must be terminated. Second is observation: physical
data must be ascertained by measurement either directly or through
other calculations. First is theory: the problem underlying the
calculation may be idealized, but further, any mathematical
formulation of a physical problem "necessarily represents only a
theory of reality, and not reality itself."

ps 4. "We may interpret the elimination method as the
combination of two tricks", von Neumann remarked as he described a
complicated algorithm by a simple relation among matrices. Analysis
and design of algorithms (many even are called matrix algorithms
today) would be impossible without this kind of simplification.
Thus it seems likely the equivalence between Gaussian elimination
and triangular factoring is a paradigm for numerical analysts, in
the sense of Thomas Kuhn. Considering its importance, its history
is too much ignored. Triangular factoring apparently originated
with T. Banachiewicz [Bull Int L'Acad Polonaise, Ser A, '38], and
later independently with P. S. Dwyer [Ann Math Stat, '44], and then
von Neumann and Goldstine, [Bull AMS, '47]. Perhaps someone can
tell Banachiewicz' story.

Anyone interested in air pollution dispersion modeling is invited to
visit this website:

http://www.air-dispersion.com

to learn about "FUNDAMENTALS OF STACK GAS DISPERSION", a comprehensive
single-source reference book on dispersion modeling of continuous,
buoyant pollution plumes. The website provides a brief description of
the book, peer reviews published in technical and scientific journals,
the book's complete table of contents, and information on how to obtain
copies.

For more details and online order, see
http://www.wkap.nl/book.htm/0-7923-4812-5

DESCRIPTION: This book presents, in a comprehensive way, the application
of optimization algorithms and heuristics in engineering problems
involving smooth and nonsmooth energy potentials. These problems arise in
real-life modeling of civil engineering and engineering mechanics
applications. Engineers will gain an insight into the theoretical
justification of their methods and will find numerous extensions of the
classical tools proposed for the treatment of novel applications with
significant practical importance. Applied mathematicians and software
developers will find a rigorous discussion of the links between applied
optimization and mechanics which will enhance the interdisciplinary
development of new methods and techniques. Among the large number of
concrete applications are unilateral frictionless, frictional or adhesive
contact problems, and problems involving complicated friction laws and
interface geometries which are treated by the application of fractal
geometry. Semi-rigid connections in civil engineering structures, a topic
recently introduced by design specification codes, complete analysis of
composites, and innovative topics on elastoplasticity, damage and optimal
design are also represented in detail.

AUDIENCE: The book will be of interest to researchers in mechanics, civil,
mechanical and aeronautical engineers, as well as applied
mathematicians. It is suitable for advanced undergraduate and graduate
courses in computational mechanics, focusing on nonlinear and nonsmooth
applications, and as a source of examples for courses in applied
optimization.

High Performance Computing Center North (see http://www.hpc2n.umu.se)
is hosting the fourth International Workshop on Applied Parallel
Computing (PARA98) in June 14-17, 1998 at Umea University, Sweden.

The general theme for PARA98 is Large Scale Scientific and
Industrial Problems focusing on:
o High-performance computing applications in academia and industry,
o Tools, languages and environments for high-performance computing,
o Scientific visualization and virtual reality applications in
academia and industry,
o Future directions in high-performance computing and networking.

The PARA98 meeting is aimed to be an international forum for
idea and competence exchange for specialists in parallel computing,
visualization, etc and scientists from industry and academia
solving large scale computational problems.
Another important aim of the PARA meetings is to strengthen the ties
between HPC centers, academia, and industry in the Nordic countries
as well as worldwide.

The meeting starts with a one day tutorial followed by a three day
workshop. There will be several invited one-hour lectures as well as
contributed 20-30 minutes talks. The conference proceedings will be
published by Springer Verlag in their LNCS series.

FIRST SUMMER SCHOOL OF THE
F O N D A P I N A P P L I E D M A T H E M A T I C S:
Numerical Analysis and Mathematical Modelling
CONCEPCION, CHILE, January 19 - 30, 1998

http://www.ing-mat.udec.cl/fondap/escuela.html

The National Foundation for Scientific and Technological Research of
Chile (CONICYT-CHILE), through its Program "FONDO NACIONAL DE AREAS
PRIORITARIAS" (F O N D A P) in APPLIED MATHEMATICS, and the Department
of Mathematical Engineering of the University of Concepcion, are very glad
to invite you to attend the FIRST SUMMER SCHOOL ON NUMERICAL ANALYSIS AND
MATHEMATICAL MODELLING, to be held at the University of Concepcion,
Concepcion, Chile, during the period January 19 - 30, 1998.

There are a limited number of fellowships (about 50) for financing
registration fees, round trip tickets and stay expenses of chilean
and foreigners students (only graduate and advanced undergraduate
students)

The 6th Annual European Symposium on Algorithms (ESA '98) will be held
in Venice, Italy, August 24--26, 1998. The Symposium covers research
in the use, design, and analysis of efficient algorithms and data
structures as it is carried out in computer science, discrete applied
mathematics and mathematical programming. Papers are solicited
describing original results in all areas of algorithmic research,
including but not limited to: Approximation Algorithms; Combinatorial
Optimization; Computational Biology; Computational Geometry; Databases
and Information Retrieval; Graph and Network Algorithms; Machine
Learning; Number Theory and Computer Algebra; On-line Algorithms;
Pattern Matching and Data Compression; Symbolic Computation. The
algorithms may be sequential, distributed or parallel, and they should
be analyzed either mathematically or by rigorous computational
experiments. Submissions that report on experimental and applied
research are especially encouraged.

Further Information: All questions should be mailed to esa98@dsi.unive.it.

Additional information may be found through anonymous FTP or
World-Wide Web:

We are planning a BLAS Technical Forum meeting on December 3-5.
The meeting will be hosted by the University of Tennessee and
will be held at the Hilton Hotel in Knoxville, TN:
Hilton Hotel
501 W Church Street
Knoxville, TN
(423) 523-2300
(800) 445-8667

The Forum has been established to consider expanding the Basic Linear
Algebra Subprograms (BLAS) in a number of directions in the light of
modern software, language, and hardware developments. The first meeting
of the Forum was held in Nashville on February 19-20, 1996 and the
previous meeting was in Portland, OR on August 14-15, 1997.

Working groups have been established to consider the overall
functionality, possible extensions, and a lightweight interface
for the BLAS, as well as the short term goals of the forum. Other
subgroups have also been established, either to advise the current
working groups or as placeholders for future working groups on parallel
processing issues, sparse operations, and language binding issues.

We strongly urge and encourage attendance at the meeting
so that we can make tangible progress towards much needed standards.
Wide input is needed to help ensure that emerging proposals are useful
and acceptable to the community.

It is appreciated that it is not easy for everyone to attend the
meetings of the Forum, but we would nevertheless welcome your input
since we wish the discussion to be as open as possible, and the results
to reflect consensus from the community at large.

We will plan to start the meeting with lunch at 12:00 pm on Wednesday,
December 3rd and end by Noon on Friday, December 5th. There will
be no registration fee, and lunch is provided on December 3 and 4.

This workshop will feature two main series of lectures by prominent
numerical analysts. There will also be scheduled lectures by other
participants. Provision will be made for impromptu presentations on
work, as it develops, arising from discussions amongst people taking
part in the workshop. The venue will be the City Campus of the
University of Auckland.

Invited speakers

The following internationally known speakers have agreed to present
series of lectures at the workshop

The Mathematical Sciences Department of Worcester Polytechnic Institute
(WPI) invites applications for anticipated tenure-track faculty positions
in applied and computational mathematics beginning in the fall of 1998.
Appointments will probably be at the Assistant Professor level, but
exceptionally well qualified candidates may be considered for appointments
at higher rank. An earned Ph.D. or equivalent degree is required. A
successful candidate must be able to contribute strongly to both the
department's research activities and its innovative, project-based
educational programs. Areas of research in the department include partial
differential equations with applications in fluid and continuum mechanics,
composite materials, computational modeling and simulation, numerical
linear algebra and nonlinear equations, optimization, control theory,
applied probability, discrete mathematics, and applied statistics.

WPI is an innovative technological university of engineering, science,
management, and the humanities and arts. It is private and highly selective,
with an enrollment of 2700 undergraduates and about 1000 full- and part-time
graduate students, and is ranked among the top 50 national universities.

The WPI campus is located in Worcester, MA, New England's second largest
city, in close proximity to the city's many cultural attractions as well
as nine other institutions of higher education in the urban area. Worcester,
forty miles west of Boston, offers access to the diverse cultural and
recreational resources of New England and provides opportunities for
urban, suburban or rural lifestyles. WPI offers a smoke-free environment.

The Mathematical Sciences Department has 23 full-time faculty and
supports a PhD program and MS programs in applied mathematics and
applied statistics, as well as a full undergraduate program. For
additional information about the Mathematical Sciences Department
and WPI, see http://www.wpi.edu/Academics/Depts/Math/.

Qualified applicants should send a detailed curriculum vitae, a one-page
statement of their specific teaching and research objectives, and the names
of four references with mail/email addresses and telephone/fax numbers to

The Department of Mathematics at Temple University invites
applications for a tenure-track position at the Assistant
Professor level, pending budgetary approval.
Applicants must have a Ph.D. and a solid record of
accomplishment or outstanding promise in research.
Priority will be given to applied mathematics, especially
scientific computing.

Temple has made a strong commitment to excellence in
undergraduate education. Candidates should have a proven record
of effectiveness in teaching mathematics at all levels.
The position will start in the fall of 1998. Salary is competitive.
Research achievements and teaching together with significant
contributions to our undergraduate programs will be the primary
criteria for advancement.

Women and minorities are particularly invited to apply.

We request that applicants send the AMS Application Cover Sheet
along with their vita and at least three letters of recommendation to:

The vita should include a description of the applicant's teaching
experience and evidence of teaching ability, and a statement
of professional goals.
The application deadline is January 16, 1998.
For further information, see
http://www.math.temple.edu/position.fold/position.html

The Engineering and Physical Sciences Research Council are funding a
Research Assistantship for a project titled
``Numerical Analysis of the Generalized Eigenvalue Problem''.
This is a three-year post, starting in January 1998,
or as soon as possible thereafter.

The successful applicant will join the Numerical Analysis group at the
University of Manchester. The aim of the project is to develop theory,
algorithms and software for the generalized eigenvalue problem.

Applicants should hold, or expect to complete before the start date,
a Ph.D., and should have a strong background in numerical analysis,
linear algebra, and software development.
The appointment will be made at the RA 1A level, with a current
starting salary of 16,927 pounds sterling per annum.

The Parallel Computing Sciences Department at Sandia National Laboratories
is seeking qualified candidates for a Post-Doctoral position to contribute to
the development of dynamic load-balancing libraries for scientific simulations
on parallel computers. Candidates must be U.S. citizens, have earned a
Ph.D. or equivalent, and have experience with the design and implementation
of dynamic load-balancing algorithms on distributed memory computers.
Experience with algorithm development on heterogeneous parallel systems
is particularly desirable. Candidates should be familiar with numerical
techniques such as adaptive finite element and difference methods. Strong
communication skills will also be needed to interact with application
developers who will use the resulting libraries. A working knowledge of
C or C++ programming and experience with software-library development
would be useful.

The Parallel Computing Sciences Department maintains research programs
in a variety of areas, including robust automatic mesh generation,
parallel algorithms, adaptive finite element methods, domain partitioning
and load balancing. Strong collaborations exist between the
Department and other departments at the Laboratories, supporting
research in mathematics and algorithms, computational physics and
engineering, and advanced systems software and tools. A unique
parallel computing environment is available, including a 4500-node
Intel TFlops computer, an 1800-processor Intel Paragon,
a 192-processor SGI Origin system, an 84-processor DEC-8400 system,
and experimental heterogeneous computer platforms.

The post-doc appointment is for a period of one year and may be
renewed for a second year. It includes a competitive salary,
reimbursement of moving expenses, and a professional travel allowance.
Applicants should send a resume and three letters of recommendation to:

A brief description is given below for a planned 3-year project. However
depending the interests of suitably qualified candidates, it is possible
to work on a different project in the general area of applied
mathematical modelling and techniques.

Title: "Fast iterative methods for adaptive solution of fluid
equations with applications to thermal contact problems"
. This project is concerned with understanding the dynamics of compressible
flows and developing effective numerical methods for solving the
discretized nonlinear systems. It involves close collaboration with
the Shell Research Company in Chester, and some financial support.
. The project, to be supervised by Dr Chen (Liverpool) and Dr Scales
(Shell), provides interesting and varied training in analytical and
numerical methods using state-of-the-art computational facilities. There
is a considerable demand from industry for Ph.D graduates with this
type of research training.

Usual EPSRC eligibility rules apply i.e. UK students - full support,
EU students - only full fees and Non-EU students - no support.

We welcome young candidates (below 30 years of age) with an M.Sc. degree in
a relevant topic and a sound background in Mathematics and experienced in
scientific programming. Research activity and/or professional experience
will additionally evaluated in favour of the candidate.

University of Texas at San Antonio
Ph.D. Computer Science Fellowships
and
M.S. and Ph.D. Research Assistantships

In an effort to recruit outstanding computer science graduate
students, the University of Texas at San Antonio (UTSA) has allocated
funds for a number of fellowships. These fellowships provide full
tuition and fees plus a stipend of $15,000. There are also a number
of graduate student research assistant positions supported by
federally funded research projects in computer systems. Particular
interests are in parallel and distributed computing systems and
applications, parallelizing compilers, high-speed networks,
performance evaluation and simulation.

The Division of Computer Science has 13 faculty and about 480
undergraduate and 60 graduate students (MS and PhD) and is expected to
grow significantly in the next few years. In addition to excellent
general purpose computing and networking facilities, the Division
supports clusters of high-performance and SMP workstations on
high-speed networks for research use. Additional information about the
academic programs and the faculty and their research can be found at
the URL http://www.cs.utsa.edu.

Applicant must have at least a BS degree in computer science or
related disciplines and should be eligible for admission as an M.S. or
Ph.D. degree seeking student in the Division of Computer Science, The
University of Texas at San Antonio.

Prospective students are requested to send a letter of interest, with
a copy of their resume to :

Kleanthis Psarris
Division of Computer Science
The University of Texas at San Antonio
San Antonio, TX 78249-0667
Email: psarris@ringer.cs.utsa.edu

Email communication is preferred. Minority and women students are
especially encouraged to apply.

Richard Sincovec, Director Telephone: 210-458-4434
Division of Computer Science Fax: 210-458-4437
The University of Texas at San Antonio e-mail: sincovec@cs.utsa.edu
6900 North Loop 1604 West
San Antonio, Texas 78249-0677